A box with a square base and open top must have a volume of 48668...
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A box with a square base and open top must have a volume of 48668 cm. We wish to find the dimensions of the box that minimize the amount of material used.First, find a formula for the surface area of the box in terms of only a, the length of one side of the square base.[Hint: use the volume formula to express the height of the box in terms of x.]Simplify your formula as much as possible.A(x) =Next, find the derivative, A'(x).A'(x) =Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x.]A'(x) = 0 when x =We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x).A"(x)=Evaluate A"(x) at the x-value you gave above.NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)
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